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<span class="kw">let </span><font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</font> be   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c2">S</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> implies <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font></span><br/>

<div><span class="lab"><font color="Green" title="E10">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g1</font> implies <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_1_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><a NAME="E2:24_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>
 ;<br/><a NAME="E3:24_1_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D10" title="BORSUK_1:def.10">BORSUK_1:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E5:24_1_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c7">y</font>,<span class="p1">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="eqrel_1.html#T51" title="EQREL_1:th.51">EQREL_1:51</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c9">ty</font> = <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>;<br/><a NAME="E7:24_1_1_1"/>
<span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a></span>;<br/><a NAME="E8:24_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></span>: 
( <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">x</font> or <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:24_1_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/><a NAME="E9:24_1_1_1"/>
 <a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D11" title="BORSUK_1:def.11">BORSUK_1:def 11</a></span>;<br/><a NAME="E10:24_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A8</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D1" title="BORSUK_1:def.1">BORSUK_1:def 1</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E11:24_1_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" title="T_1TOPSP:func.3">T_1-reflect</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24_1_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E8:24_1_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>
<br/>.= 
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E10:24_1_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E11:24_1_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:24_1_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:24_1_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:24_1_1"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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